Unit Vector Calculator

In this article, the calculator developed by us will easily calculate the unit vector that is the vector of length 1 unit in any plane. The article will first explain to you the concept of the unit vector in physics, its formulas followed by a few solved examples and generally asked FAQs. 

A unit vector is a vector defined that of length equal to 1unit in any plane. When we use a unit vector to describe a spatial direction, we call it a direction vector with respect to that plane. In a Cartesian coordinate system(which we generally work in), the three-unit vectors that form the basis of the 3D space are as follow:

(1, 0, 0) — Directs the vector in the x-direction;

(0, 1, 0) — Directs the vector in  the y-direction; and

(0, 0, 1) — Directs the vector in the z-direction.

What is a unit vector?

Talking about an arbitrary vector, it is possible to calculate what the unit vector is along the same direction in any plane. And to do that, we can apply the following formula to calculate the unit vector ; û = u / |u|


û — Unit vector;

u — Arbitrary vector in the form (x, y, z) or in any dimensional plane; and

|u| — Magnitude of the vector u in one particular direction.

You can calculate the magnitude of a vector using our distance calculator or simply by the equation:

|u| = √(x² + y² + z²)

The unit vector is a useful concept in describing linear transformations of any vector in a plane. For example, the matrix norm describes how much a unit vector is stretched when multiplied by a matrix of a dimensional vector.

How to calculate the unit vector?

Let's consider an example of a vector u = (1, -3, 2). To calculate the unit vector in the same direction, you have to follow these steps:

  1. Write down the x, y, and z components of the vector. In this case, x₁ = 1, y₁ = -3 and z₁ = 2.
  2. Calculate the magnitude of the vector u:
    |u| = √(x₁² + y₁² + z₁²)
    |u| = √(1² + (-3)² + 2²)
    |u| = √(1 + 9 + 4)
    |u| = √14
    |u| = 3.8
  3. Now that you know the magnitude of the vector u, you probably want to know how to calculate the unit vector. All you have to do is divide each of the initial vector's components by |u|.
    x₂ = x₁ / |u| = 1 / 3.8 = 0.263
    y₂ = y₁ / |u| = -3 / 3.8 = 0.789
    z₂ = z₁ / |u| = 4 / 3.8 = 1.05
  4. Now, write these results in vector form to find the vector û = (0.263,0.789,1.05).
  5. The user can verify the result if the magnitude of your unit vector should be equal to 1.

Solved Examples on Unit Vector

Question 1:
Find the unit vector →aa→for the given vector, 12ii^– 3^jj^– 4 ^kk^. 



To find the magnitude of the given vector first, →aa→ is :

Let’s use this magnitude to find the unit vector now:

Substituting the values:



The unit vector in Bracket form is given as follow:


Question 2: Find the unit vector →bb→ for the given vector, −2^i+4^j–4^k.−2i^+4j^–4k^.



To find the magnitude of the given vector first, 

→b is :|b|= √x2+y2+z2|q|=√−22+(4)2+(−4)2

|bI= √4+16+16

 Ib|= √36

 |b|= 6

The magnitude to find the unit vector now:

 |b|= 6

FAQs on Unit Vector Calculator

1. How to find a unit vector in the same direction as that of the plane?

To find this, divide the original vector by its magnitude. For example, the vector u = (2, 3) has a magnitude of √(2² + 3²) = √13. Therefore the unit vector that has the same direction is û = (2/√13, 3/√13) = (0.5547, 0.832).

2. Is (1, 1) a unit vector?

No, the length of a unit vector needs to be equal to 1. Calculating the length of (1, 1), we find that √(1² + 1²) = √2 = 1.414, which is not equal to 1. Hence this is not a unit vector.

3. What is the magnitude of a unit vector?

The magnitude, or length, of a unit vector, is 1 units.

4. What is unit vector notation?

The notation you use to denote a unit vector to place a circumflex, or "hat" above the lowercase letter representing the vector. 

5. How do you find the unit vector?
To find a unit vector with the same direction as a given vector, simply divide the vector by its magnitude.